If `sin theta = 3/4 and tan theta =9/2,` then `cos theta` is

Options

A. `1/6`
B. `8/27`
C. `27/8`
D. `15/4`

Answer

To find \( \cos \theta \) given that \( \sin \theta = \frac{3}{4} \) and \( \tan \theta = \frac{9}{2} \), we can follow these steps:

### Step 1: Use the definition of tangent
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Given \( \tan \theta = \frac{9}{2} \) and \( \sin \theta = \frac{3}{4} \), we can substitute these values into the equation:
\[ \frac{9}{2} = \frac{\frac{3}{4}}{\cos \theta} \]

### Step 2: Cross-multiply to solve for \( \cos \theta \)
Cross-multiplying gives us:
\[ 9 \cdot \cos \theta = \frac{3}{4} \cdot 2 \]
This simplifies to:
\[ 9 \cdot \cos \theta = \frac{3 \cdot 2}{4} \]
\[ 9 \cdot \cos \theta = \frac{6}{4} \]
\[ 9 \cdot \cos \theta = \frac{3}{2} \]

### Step 3: Isolate \( \cos \theta \)
Now, divide both sides by 9:
\[ \cos \theta = \frac{3/2}{9} \]
This can be simplified as follows:
\[ \cos \theta = \frac{3}{2 \cdot 9} = \frac{3}{18} = \frac{1}{6} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{1}{6} \]

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